Elastography based on x-ray computed tomography and sound wave integration

ABSTRACT

Systems and methods for integrating a three-dimensional X-ray computed tomography system with an independent sound wave system to determine mechanical properties of tissue using signals from the sound wave system. Methods are disclosed that generate a numerical simulation and take the transmitted wave signals as the optimization objective to estimate modulus distribution of the tissue. Further, the mechanical properties of the tissue are reconstructed based on an inverse algorithm.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a non-provisional and claims benefit of U.S.Provisional Application No. 62/657,392, filed Apr. 13, 2018, thespecification(s) of which is/are incorporated herein in their entiretyby reference.

GOVERNMENT SUPPORT

This invention was made with government support under Grant No.CMMI-1229405 awarded by the National Science Foundation. The governmenthas certain rights in the invention.

BACKGROUND OF THE INVENTION Field of the Invention

The present invention relates to medical imaging and materialscharacterization, in particular, to elastography based on X-ray computedtomography (CT) and sound wave integration.

Background Art

Elastography framework relies on external excitations or perturbations,such as mechanical waves, which generate mechanical responses such aswave velocity, wavelength, and/or displacement measurable deep inside atissue to map the distribution of elastic moduli within a region ofinterest of the tissue. However, direct measurements based on imagessuffer from issues such as image noise and low spatial resolutionbecause of tissue interfaces. These problems restrict detectable tissuecategories and clinical application. Previous elastography based oncomputed tomography (CT) alone requires two CT scans, which exposes thesample to higher amounts of radiation. Ultrasound implements highfrequencies which only carry little mechanical energy and do not longdistances because of high attenuation, therefore a mechanical responsein unobtainable from the sample.

To resolve these problems, the present invention discloses anelastography method based on integration of X-ray computed tomography(CT) imaging and sound wave testing methods to characterize mechanicalproperties of tissues, without relying on the direct measurement ofinternal responses such as the wave velocity, wavelength, and/ordisplacement within the tissues. The present invention may beadditionally used to characterize mechanical properties of engineeringmaterials.

BRIEF SUMMARY OF THE INVENTION

An integrated elastography method is disclosed that combinesthree-dimensional X-ray tomography and independent sound wave test. Toput the combination into numerical simulation and take the transmittedwave signals as the optimization objective, the modulus distribution isreconstructed based on the inverse algorithm. For implementing thisinvention, the 3D image model is obtained from an X-ray CT modality, andthe excitation and the response are pulsed and collected by soundtransducers around the object, respectively. Some of transducers aresound sources, while others are receivers which are used to measure thetransmitted signals. The collected data can be displacement, velocity,acceleration or pressure on the object surface.

In some aspects, the present invention discloses an integratedelastography system for estimating mechanical properties of a sample,the system comprising an x-ray computed tomography (“CT”) systemcomprising an x-ray source and a detector, wherein the x-ray sourceemits x-rays towards the sample and wherein the detector collects x-raysemitted from the sample. The system additionally comprises a sound wavesystem having one or more acoustic transducers and one or more receiverspositioned around the outer surface of the sample, wherein the one ormore transducers are configured to generate sound waves that impinge onthe sample, and wherein the one or more receivers receive a first set oftransmitted signals, wherein the first set of transmitted signalsinclude the sound waves that transmit through the sample.

The system additionally includes a controller operably coupled to thex-ray CT system and the sound wave system, wherein the controller hasmemory that stores computer readable instructions that, when executed bythe controller, causes the controller to generate an x-ray CT image ofthe sample, using the x-ray CT system, and select a matrix and aninclusion based on geometries detected from the CT image of the sample.The controller may additionally generate the first set of transmittedsignals based on the CT image using the sound wave system, select aninitial set of values comprising one or more of an elasticity modulus, aPoisson ratio, and a density for the matrix and the inclusion based onthe selection, and perform and repeat a forward analysis sequence tosimulate a sound wave transmission through the matrix and the inclusionto deduce a mechanical response of the matrix and the inclusion. Theforward analysis sequence may include optimizing the initial set ofvalues until the set of signals generated converges with the first setof transmitted signals and the mechanical response may be deduced basedon the set of signals, thereby decoding the mechanical response withoutrelying on the response acquisition within the sample.

In some embodiments, the forward analysis sequence may includeconstructing a finite element (“FE”) model by generating a mesh of eachof the matrix and of the inclusion from the CT image, wherein the meshis generated using an approximation of the geometries of the matrix andthe inclusion, and wherein generation of the mesh includes dividing theCT image into smaller finite elements. In some embodiments, the memorymay include additional instructions that, when executed by thecontroller, cause the controller to perform an inverse analysis on theset of signals to deduce a modulus of each of the matrix and theinclusion. The inverse analysis may include a wavelet synchro-squeezedtransform (“WSST”).

In some embodiments, the WSST may include generating a continuouswavelet transform from the set of signals, extracting instantaneousfrequency from the continuous wavelet transform, reconstructingtransmitted signals from continuous wavelet transform and the frequency,and decoding the reconstructed transmitted signals into shear signalsand pressure signals, and estimating the modulus of the sample based onone or more of the shear signals and the pressure signals, therebydecoding the mechanical property of the sample using the x-ray CT imageand the transmitted signals.

In some embodiments, the estimation may be based on a time reversaltheory, wherein the modulus may be estimated from the shear signals andthe pressure signals. The one or more transducers may be disposed on asurface of an enclosure of the integrated elastography system andwherein the plurality of receivers may be in contact with the sample. Insome embodiments, the sound waves may be harmonic waves, sinusoidal toneburst, square waves, triangle waves and so on in the wide frequencyrange, such as from 100 Hz to 100 kHz. The initial set of values may beselected based on a reasonable and educated estimate of the elasticitymodulus, the Poisson ratio, and the density for the matrix and theinclusion. The mechanical response may include one or more ofdisplacement, velocity, acceleration, and a pressure within the sample.

In some embodiments, the sample may be a biological sample and thematrix may include tissue and the inclusion may include non-tissuebiological material. In some embodiments, the biological material may bea tumor or mass. In some embodiments, the sample may include particlesembedded in a solidified medium.

According to some embodiments, an integrated elastography system isprovided. The system may include an x-ray computed tomography (“CT”)system comprising an x-ray source and a detector wherein the x-raysource emits x-rays towards a sample and wherein the detector collectsx-rays emitted from the sample. The system may additionally include asound wave system having a plurality of acoustic transducers and aplurality of receivers positioned around the outer surface of thesample. The plurality of transducers is configured to generate soundwaves that impinge on the sample, and the plurality of receiversreceives a first set of transmitted signals. The first set oftransmitted signals includes the sound waves that transmit through thesample.

The system may additionally include a controller operably coupled to thex-ray CT system and the sound wave system. The controller can have amemory that stores computer readable instructions that, when executed bythe controller, causes the controller to perform operations. Theseoperations may comprise: (i) generate an x-ray CT image of the sampleand select a matrix and an inclusion based on geometries detected fromthe CT image of the sample, (ii) generate a mesh of each of the matrixand of the inclusion, wherein the mesh is generated using anapproximation of the geometries of the matrix and the inclusion, whereinthe mesh is generated from the CT image, and wherein generation of themesh includes dividing the CT image into smaller finite elements, (iii)choose an initial set of values comprising one or more of an elasticitymodulus, a Poisson ratio, and a density for the matrix and the inclusionbased on the selection, (iv) perform a finite element analysis on the CTimage using the set of initial values of an elasticity modulus, aPoisson ratio, and a density of the sample to generate a second set oftransmitted signals, (v) perform a wavelet synchro-squeezed transform(“WSST”) of a deformation of shear and pressure waves outputted fromfinite element analysis to generate a continuous wavelet transform, (vi)extract instantaneous frequency from the continuous wavelet transform,(vii) reconstruct the transmitted signals from continuous wavelettransform and the frequency, and (viii) decode the reconstructedtransmitted signals into shear signals and pressure signals. Thecontroller may repeat step (iv)-(viii) to optimize the set of initialvalues and generate the second set of transmitted signals until thesecond set of transmitted signals converge with the first set oftransmitted signals, and (ix) estimate an elastic modulus of the samplebased on one or more of the shear signals and the pressure signals,wherein the estimation is based on a time reversal theory, and wherein amechanical property of the sample is estimated from the elastic modulus,thereby decoding the mechanical property of the sample using the x-rayCT image and the transmitted signals without relying on the responseacquisition within the sample.

In some embodiments, the time reversal theory may include generating atime reversal displacement field and a time reversal strain field,wherein the elastic modulus is determined from a ratio of the timereversal displacement field and the time reversal strain field. In someembodiments, the sound waves may include a sine tone burst and theinitial set of values may be chosen based on a reasonable and educatedestimate of the elasticity modulus, the Poisson ratio, and the densityfor the matrix and the inclusion. The time reversal analysis may includea WSST.

According to some embodiments, a method for determining a modulus of asample is provided. The method may comprise obtaining an x-ray CT imageof the sample using x-rays collected from the sample. The x-rays arecollected using an x-ray CT system comprising an x-ray source that emitsx-rays towards the sample and a detector that collects x-rays emittedfrom the sample. The method continues with positioning at or near anouter surface of the sample a sound wave system comprising one or moreacoustic transducers and one or more receivers. The one or moretransducers can generate sound waves such as, for example, a sine toneburst, and the one or more receivers receive a first set of transmittedsignals. The method may additionally include impinging the sample withthe sound waves generated from the one or more transducers. The one ormore receivers then receive the first set of transmitted signals, whichinclude the sound waves that transmit through the sample. The method mayadditionally perform a finite element analysis on the CT image using aset of initial values of an elasticity modulus, a Poisson ratio, and adensity of the sample to generate a second set of transmitted signals,and perform a cycle operation of optimizing the set of initial valuesand generating the second set of signals until the second set of signalsconverge with the first set of transmitted signals; and determiningmechanical response of the sample based on the optimized set of initialvalues, wherein the mechanical response includes one or more of adisplacement, a velocity, an acceleration, and a pressure of the sample.

In some embodiments, the method may further include performing aninverse analysis on the second set of signals to deduce a modulus of thesample. The inverse analysis may include a wavelet synchro-squeezedtransform (“WSST”). The WSST may include generating a continuous wavelettransform from the set of signals, extracting instantaneous frequencyfrom the continuous wavelet transform, reconstructing transmittedsignals from continuous wavelet transform and the frequency; decodingthe reconstructed transmitted signals into shear signals and pressuresignals; and estimate the elastic modulus of the sample based on one ormore of the shear signals and the pressure signals, wherein theestimation is based on a time reversal theory, and wherein the elasticmodulus is estimated from the shear signals and the pressure signals,thereby decoding mechanical property of the sample using the x-ray CTimage and the transmitted signals.

In some embodiments, performing the finite element analysis on the CTimage may include selecting a matrix and an inclusion based ongeometries detected from the CT image of the sample; and generating amesh of each of the matrix and of the inclusion, wherein the mesh isgenerated using an approximation of the geometries of the matrix and theinclusion, wherein the mesh is generated from the CT image, and whereingeneration of the mesh includes dividing the CT image into smallerfinite elements. The method may further include determining themechanical response and the modulus of each of the matrix and theinclusion.

One of the unique and inventive technical features of the presentinvention is the transmission of sound signals entering from the outsideof the sample. Without wishing to limit the invention to any theory ormechanism, it is believed that the technical feature of the presentinvention advantageously provides for a method to determine themechanical parameters, such as pressure, velocity, displacement, strainor wavelength, from the outside of the sample and obtain the physicalproperties of the sample from the parameters. This eliminates that needto use complex and/or invasive imaging techniques to acquire mechanicalparameters inside the sample. None of the presently known priorreferences or work has the unique inventive technical feature of thepresent invention.

Moreover, waves in solid have mode conversion and scattering atinterfaces, which introduces challenges in modeling and inverseconvergence. Traditionally, in order to inversely have a converging andreliable analysis, internal measurement of mechanical responses isnecessary. Thus, one of ordinary skill in the art would perceive thatthe combination of using x-ray CT imaging with acoustic sound would notwork because information from the outside of the sample would beinsufficient. Contrary to current belief, the inventive technicalfeature of using x-ray CT imaging with acoustic sound at the outside ofthe sample was surprisingly found to work in determining the mechanicalproperties of the sample. This provided an additional benefit ofeliminating the need for multiple CT scans, thus only one x-ray CT imageof the sample was required to perform elastography.

Any feature or combination of features described herein are includedwithin the scope of the present invention provided that the featuresincluded in any such combination are not mutually inconsistent as willbe apparent from the context, this specification, and the knowledge ofone of ordinary skill in the art. Additional advantages and aspects ofthe present invention are apparent in the following detailed descriptionand claims.

BRIEF DESCRIPTION OF THE DRAWINGS

This patent application contains at least one drawing executed in color.Copies of this patent or patent application publication with colordrawing(s) will be provided by the Office upon request and payment ofthe necessary fee.

The features and advantages of the present invention will becomeapparent from a consideration of the following detailed descriptionpresented in connection with the accompanying drawings in which:

FIG. 1A shows a non-limiting embodiment of an integrated elastographysystem of the present invention.

FIG. 1B shows a schematic diagram of the integrated elastography systemhaving a controller and an X-ray computed tomography (CT) integratedwith a sound wave system.

FIG. 2 is a flow chart depicting an example elastography method thatintegrates X-ray CT imaging and sound wave testing.

FIG. 3 shows a schematic illustration of sound waves incident on asample positioned within the integrated elastography system, andtransmitted signals collected using receivers of the integratedelastography system.

FIG. 4A shows a schematic of a two-dimensional (2D) model and across-section of a three-dimensional (3D) model used for finite element(FE) simulation.

FIG. 4B shows a non-limiting example of a pulse input used for the FEsimulation, where t1=t2=10⁻⁴ s.

FIGS. 5A-5D show a pressure distribution of the 2D used for FEsimulation.

FIGS. 6A-6B show pressure distributions of the 3D model used for FEsimulation.

FIG. 7 shows a numerical composite material model.

FIG. 8 shows a wavelet synchro-squeezed transform (WSST) signalprocessing.

FIG. 9A shows a picture of a sample made of silicone matrix having aneraser inclusion, and FIG. 9B shows a cross-section of the sample ofFIG. 9A.

FIG. 10 shows a 3D image of the inclusion.

FIG. 11 shows a profile of an incident wave transmitting through thesample.

FIG. 12 shows a FE simulating model.

FIG. 13 shows plots of objective average of each level and a levelfactor for each factor.

FIG. 14 shows FE pressure output of optimum values.

FIG. 15 shows pictures of the samples used for compression tests.

FIGS. 16A-16B shows results of the compression tests of the siliconematrix and the eraser inclusion.

FIG. 17 shows a brain-tissue slice segmented into four regions meshedfor FEM simulation, including the cortex (C), corona radiata (CR),corpus callosum (CC) and basal ganglia (BG). The four arrows show theincidence positions and the 12 black solid dots show the signaldetection locations. The ‘Fixation’ boundaries marked by red curves meanthat displacement is constrained to be zero during FEM simulation.

FIGS. 18A-18B show a comparison of the target signals and the best trialsignals in the first round and the fourth round, respectively. ‘Ob’stands for the target signals and ‘Tr’ for the output ones from the besttrial.

DESCRIPTION OF PREFERRED EMBODIMENTS

Following is a list of elements corresponding to a particular elementreferred to herein:

100 integrated elastography system 101 x-ray CT imaging system 102 x-raysource 103 sound wave system 104 objective 108 detector 110 sample 112transducer 114 receiver 120 controller 202 incident waves 204 object

As used herein, the term “matrix” corresponds to a substrate, which maybe of a continuous material. An “inclusion” refers to a material(s),i.e. object, embedded in the matrix. For instance, in biologicalapplications, the matrix may include a tissue and the inclusion may be atumor. Matrix and inclusion represent different tissues/materials. Thematrix is the continuous material into which the inclusion(s) is/areembedded, whereas the inclusion is the material embedded into thematrix. In engineering, particle-filled composites are examples ofmatrix/inclusion systems.

As used herein, the term “mesh” refers to a representation of finiteelement. The mesh is generated by dividing the matrix and inclusion intosmaller finite elements.

Referring now to FIGS. 1-18B, the present invention features anintegrated elastography method that combines 3D X-ray CT and independentsound wave testing methodologies. More specifically, the presentinvention includes performing numerical simulation using transmittedsound wave signals as the optimization objective and reconstructing amodulus distribution based on an inverse algorithm. A non-limitingembodiment of the integrated elastography system (100) of the presentinvention is shown in FIG. 1.

In one embodiment, the integrated elastography system (100) may includea 3D x-ray CT imaging system (101) having an x-ray source (102) and adetector (108). A sample (110) may be positioned in the integratedelastography system (100) such that x-rays emitting from the source(102) impinge on the sample, and the transmitted x-rays may be focusedby an objective (104) onto the detector (108). A controller (120) of theintegrated elastography system (100) may generate a 3D image of thesample from the signals received by the detector (108). The integratedelastography system (100) may additionally include a sound wave system(103) having a plurality of transducers positioned around the sample forexciting the sample and a plurality of receivers positioned around thesample for receiving the response of the sample due to the excitation.As an example, the plurality of transducers may include acoustic orsound transducers (112) configured to generate sound waves, andadditionally include acoustic receivers (114) configured to receivesignals transmitted through the sample.

For illustrative purposes, only one acoustic transducer and one acousticreceiver is shown in FIG. 1A. However, it is to be understood that thenumber of transducers and receivers should be enough for exciting eachphase and receiving the transmission and reflection through each phase,for acquiring the mechanical responses outside the sample withoutinternal measurement. In some embodiments, the number of transducers maybe 1-5 or greater than 5. In other embodiments, the number of receiversmay be 1-5 or greater than 5. In yet other embodiments, the number ofreceivers may be greater than the number of transducers. For instance, atransducer may emit the sound waves, and multiple receivers may bepositioned around the sample to collect the signals transmitting thoughthe sample as shown in FIG. 3.

In non-limiting examples, a plurality of acoustic transducers and aplurality of receivers may be positioned around the outer surface of thesample. In some embodiments, an acoustic transducer, such as a pulser,may be assembled on a frame attached to a roof of an enclosure of theintegrated elastography system (100) and a receiver may be positionedsuch that the receiver is in contact or touches the outer surface of thesample (110). Herein, incident waves (204) may be emitted from theacoustic transducer (112), and the sound waves (204) may pass throughthe sample (110) and an object (206) within the sample (110). Signalstransmitted from both the sample and the object may be collected bymultiple receivers (114) positioned around the sample (110), forexample, near or at the sample's surface. The controller (120) may usethe transmitted signals collected from the acoustic receivers (114) togenerate information such as displacement, velocity, acceleration orpressure on the sample surface, for example. The controller (120) of theintegrated elastography system (100) may control one or more of thecomponents of the x-ray CT imaging system (101) and the sound wavesystem (103).

According to another embodiment, the present invention discloses anelastography method (1700) that integrates X-ray CT and sound wavetesting to estimate the mechanical properties from transmitted soundsignals, without using any complex image processing techniques. Turningnow to FIG. 2, at 1702 of method 1700, a 3D x-ray CT scanning of anobject/sample/tissue positioned within an integrated elastography system(such as the integrated elastography system (100) of FIG. 1A) isperformed. Using an x-ray source of the integrated elastography system,an x-ray CT image of the objects is generated. At 1704, method 1700includes performing a sound test on the object wherein the sound testincludes exciting the object with an incident wave emitted from aplurality of sources and receiving a first set of transmitted signalspassing through the object using a plurality of receivers. Preferably,the plurality of sources and the plurality of receivers are positionedat or near an outer surface of the sample (110).

In some embodiments, the present invention utilizes two procedures: aforward analysis (e.g. a finite element analysis (performed at 1706 and1708)) to estimate a mechanical response of the object, and an inverseanalysis (e.g., a wavelet synchro-squeezed transform (WSST) (performedat 1710 and 1712)) to estimate a mechanical property of the object, asexplained below.

For the forward analysis procedure, the method proceeds to step 1706. Inone embodiment, the method includes constructing an FE model to simulatea second set of transmitted signals by generating a mesh of the x-ray CTimage (obtained at 1702) and further setting up a loading conditionbased on incident waves. As an example, the mesh may be generated basedon the CT image and the wavelength. The mesh type may be same for boththe matrix and the inclusion. The loading condition may be the externalexcitation in FE simulation and set and copied as the incident wave at1706.

The FE model may be continuously generated by changing an initial set ofvalues so that the second set of transmitted signals converges with thefirst set of transmitted signals, for example. Herein, the initial setof values includes one or more of an elasticity modulus, a Poissonratio, and a density for the matrix and the inclusion based on theselection. As such, steps 1706 and 1708 form a calculating cycle inwhich 1708 provides guessing a set of values for 1706. The cycle comesto end when the FE simulation meets the measurements. To summarize, withan educated guess of the initial set of values of matrix and inclusion,the FE simulation starts. FE output (or the second set of transmittedsignals) is determined based on the initial set of values. At eachiteration of the FE model, the set of values such as the elasticitymodulus, the Poisson ratio, and the density for the matrix and theinclusion are changed and the output of the FE model is generated usingthe new set of values. After the iteration is completed, its output iscompared with the real measurement (e.g., first set of signals). If theFE output matches with the real measurement, the FE simulation stops.Otherwise, FE simulation continues with newly estimated mechanicalparameters of the matrix and inclusion. The FE model is explained indetailed below, using numerical simulation.

SIMULATION EXAMPLE 1

The following is a non-limiting example of a numerical simulation. It isto be understood that the example described herein is presented forillustrative purposes, and is not intended to limit the invention in anyway. Equivalents or substitutes are within the scope of the invention.

Numerical Simulation

The finite element analysis (FEA) is a numerical method for solvingproblems of engineering and mathematical physics. FEA formulation of theproblem results in a system of algebraic equations. The method yieldsapproximate values of the unknowns at discrete number of points over thedomain. To solve the problem, a large problem is subdivided into smallerparts that are called finite elements. The equations that model thesefinite elements are then assembled into a larger system of equationsthat models the entire problem. FEM then uses variational methods fromthe calculus of variations to approximate a solution by minimizing anassociated error function.

In order to establish the working of the FEA for the elastographysystem, instead of an actual CT scan, a 2D model is used. The 2D modelshown in FIG. 4A includes an air domain (350×350 mm), a matrix (50×150mm), and an inclusion (R10 mm). It may be appreciated that the tissuemay be referred to as the matrix and organelles within the tissue may bereferred to as the inclusion. As an example, the model may be consideredfor breast cancer (organelle) in breast tissues (matrix). As anotherexample, aggregate (organelle) in concrete mortar (matrix) may berepresented by the 2D model. Assumed properties are the elastic modulusof 1.5 kPa and the density of 1050 kg/m3 for the matrix, and the elasticmodulus of 15 kPa and the density of 1050 kg/m3 for the inclusion. Assuch, the assumed properties are the first inputs that are used foroptimization analysis and are typically obtained with reasonable guess.Matrix and inclusion are treated as linearly elastic and are assumed tohave the same Poisson ratio of 0.49. A non-limiting example of anincident pulse wave is shown in FIG. 4B. The 3D simulation is modeled asthe way expanding the 2D geometry in depth direction, 350×350×350 mm airdomain, 50×50×150 mm matrix and spherical inclusion of 10 mm radius.Therefore, the 3D model has same cross-section across the center of theinclusion as the 2D model shown in FIG. 4A. Meanwhile, the 3D model hasthe same material properties and incident wave as those of the 2D model.

In both the 2D and the 3D numerical cases, the optimizing variables arethe moduli of matrix and inclusion. In some embodiments, full factorialexperiments may be employed to do the optimization instead of findingexplicit solutions between the transmitted pressure and elastic moduliof samples. At this point, levels for each factor (variable) should beset firstly for using full factorial experiments method. In the twocases, 5 levels for each factor are designed: [0.8 1.2 1.6 2.0 2.5] kPafor the elastic modulus of matrix and [10 13 15 22 25] kPa for themodulus of inclusion. According to the principle of full factorialdesign, each case of 2D and 3D has 25 tries (25 times simulation). Thetypical pressure distribution of the 2D model is shown in FIG. 5, andthat of the 3D model is shown in FIG. 6.

According to all of simulation and range analysis, the optimal value ofthe elastic modulus of the matrix for both 2D and 3D, 15 kPa, can bedirectly optimized out. The optimal value of the inclusion can beidentified by parametric analysis. As such, the parametric analysisincludes examining the behavior of the outputs as one or more of theinputs (or parameters) are systematically varied.

It can be concluded from these two numerical cases that the modulus canbe identified in light of transmitted waves. At this point, theintegration method has been initially validated with FE simulation.

In some embodiments, the wavelet synchro-squeezed transform (WSST) maybe used on the output of FE method to estimate a mechanical propertysuch as modulus of the matrix and the inclusion. The WSST may be used todecode pure shear or pure pressure waves and calculating the mechanicalproperty based on the time reversal theory, as explained below.

Wavelet Synchro-Squeezed Transform (WSST) Method

The procedure using the wavelet synchro-squeezed transform (WSST) startsfrom the continuous wavelet transform (CWT),

$\begin{matrix}{{X_{f}\left( {a,b} \right)} = {\frac{1}{\sqrt{a}}{\overset{\infty}{\int\limits_{- \infty}}{{x(t)}{\overset{\_}{\varphi}\left( \frac{t - b}{a} \right)}{dt}}}}} & (1)\end{matrix}$where x(t) is the original signal, ϕ the complex conjugate of the motherwavelet, a the scale variable, and b the shifted variable. The next isto extract the instantaneous frequencies, ω_(f) (a,b), from X_(f) (a,b),

$\begin{matrix}{{\omega_{f}\left( {a,b} \right)} = {{- \frac{i}{X_{f}\left( {a,b} \right)}}\frac{\partial{X_{f}\left( {a,b} \right)}}{\partial b}}} & (2)\end{matrix}$

Then WSST is the following,

$\begin{matrix}{{T_{f}\left( {\omega_{l},b} \right)} = {\frac{1}{\Delta\omega}{\sum\limits_{a_{k}:{{{{\omega_{f}{({a_{k},b})}} - \omega_{l}}} \leq {{\Delta\omega}/2}}}{{X_{f}\left( {a_{k},b} \right)}{a_{k}^{- \frac{3}{2}}\left( {\Delta\; a} \right)}_{k}}}}} & (3)\end{matrix}$where ω_(l) is the central frequency.

Accordingly, the original signal can be reconstructed as follows,

$\begin{matrix}{{x(t)} = {{Re}\left\lbrack {C_{\varphi}^{- 1}{\sum\limits_{l}{{T_{f}\left( {\omega_{l},b} \right)}{\Delta\omega}}}} \right\rbrack}} & (4)\end{matrix}$where

${C_{\varphi} = {\frac{1}{2}{\overset{\infty}{\int\limits_{0}}{{\hat{\varphi}(ɛ)}\frac{d\; ɛ}{ɛ}}}}},$with ϕ{circumflex over ( )}(ε) being the Fourier transform of the motherwavelet.

The mother wavelet is commonly the Morlet wavelet,

$\begin{matrix}{{\varphi(t)} = {\sqrt{\frac{2}{\pi\; f_{l}}}\left( {e^{{i2}\;\pi\; f_{lt}} - e^{{- \frac{1}{2}}{({2\pi\; f_{l}})}^{2}}} \right)e^{{- \frac{1}{2}}t^{2}}}} & (5)\end{matrix}$

WSST is used to decode what signals are the shear field and what are thepressure field. After that, the time reversal field will step in asfollows,D ^(TR)(r,t)=d(r ₀ ,−t)⊗d(r,t)=Σ_(τ) d(r ₀,τ)d(r,t+τ),  (6)ϵ^(TR)(t,t)=ε(r ₀ ,−t)⊗ε(r,t)=Σ_(τ)ε(r ₀,τ)ε(r,t+τ),  (7)where ⊗ is the convolution operator, D^(TR)(r,t) the time reversaldisplacement field, d(r,t) the measured displacement field, d(r₀,−t) thetime-reversed displacement field of a virtual point sources at r₀,ϵ^(TR) (r,t) the time reversal strain field, ε(r,t) the measured strainfield, ε(r₀,−t) the time-reversed strain field of a virtual pointsources at r₀. With these time reversal fields, if they are the pureshear or pressure wave induced deformation, the wavelength, λ(r₀), is:

$\begin{matrix}{{\lambda\left( r_{0} \right)} \approx {\frac{D^{TR}\left( {r_{0,}0} \right)}{\epsilon^{TR}\left( {r_{0},0} \right)}.}} & (8)\end{matrix}$

The wave velocity and the elastic modulus can be figured out in terms ofλ(r₀). Thus, a new method of characterizing the material's mechanicalproperties has been provided.

In order to verify the method, as shown in FIG. 7 (406 mm×76 mm×102 mm),a 3D composite model is built in which the volume fraction of the matrixis 60% and that of the inclusion is 40%. The elastic modulus, Poissonratio, and the density are 14 GPa, 0.17 and 2197.5 kg/m³, respectively,for the matrix and 45 GPa, 0.17 and 2688.9 kg/m³ for the inclusion.

During numerical simulation, a pulse wave in FIG. 4 is applied at themiddle of the surface. Then the induced displacement and strain fieldscan be obtained. Two points near the two ends are picked for analysis.FIG. 8 shows a typical signal process using WSST, in which thereconstruction of WSST processing on the top left corner shows apparentshear signal that will be used to calculate the modulus based onequation (6) through equation (8). Finally, the elastic modulus is 16.0GPa at one end and 20.9 GPa at another end, which illustrates the effectof composite heterogeneity. The new method applies to any frequencyrange. In this way, the mechanical properties of the tissue may beestimated using the WSST model using the transmitted signals and withoutusing complex image processing. Thus, method 1700 requires transmittedsignals to generate the mechanical property of the object either usingFE model or using WSST model. It may be appreciated that no data isneeded from inside of the object, resulting in an easier to implementmethod which is unaffected by noise in the imaging system. In addition,the method is more efficient because the analysis of the mechanicalproperties is performed without using any sophisticated imageprocessing.

While it has been described a forward analysis using FE methods, and aninverse analysis using WSST and time reversal theory, it is to beunderstood that the present invention in not limited to the use of thesespecific methods and techniques. In some embodiments, the presentinvention may utilize any suitable and appropriate forward analysismethods and inverse analysis methods known to one of ordinary skill inthe art.

Experimental Validation

The present invention has been validated both in numerical simulationstage and experimental stage.

Sample: The sample is made of the silicone rubber as the matrix and theeraser material as the inclusion as shown in FIG. 9. The first step isto scan the sample to get the internal structure of the sample. FIG. 10shows the inclusion geometry scanned by Nano CT. The second step to havesound test on the same sample and acquire the transmitted signals. Theincident wave in current experiments is a 30 kHz sine tone burst of 4467Pa peak value as shown in FIG. 11. The output peak is 22.4 Pa that willbe optimization objective. The pulser and receiver have the sameaperture, 17 mm. The third step includes FE modeling wherein theinclusion geometry resolved by Nano CT scan is imported into COMSOL,which is a commercial FE software as shown in FIG. 12 in which matrixgeometry is built by COMSOL. One smaller circle on the top of the sampledemonstrates the loading area of the incident wave, another one below onthe bottom demonstrates the transmitted signal acquisition area wherethe receiver is placed. The two circles have the diameter of 17 mm sameas the aperture of sound transducers. The fourth step includesperforming inverse analysis. In this experimental case, 6 factors(optimizing variables) are chosen: moduli, Poisson ratio and density ofmatrix and inclusion, 5 levels are set for each factor as presented inTable 1. Because there are more factors, the orthogonal array design isemployed to conduct the inverse analysis. The array of six 5-levelfactors is totally 25 times simulation according to the principle oforthogonal design.

TABLE 1 Levels for each factor is shown below. Material Factors LevelsMatrix Modulus 1.2 1.6 2.0 2.4 2.8 (MPa) Poisson 0.30 0.35 0.40 0.450.49 Ratio Density 1000 1050 1100 1150 1200 (kg/m³) Inclusion Modulus2.2 2.6 3.0 3.4 3.8 (MPa) Poisson 0.35 0.39 0.43 0.47 0.49 Ratio Density1500 1600 1700 1800 1900 (kg/m³)

In order to reduce computational time, the transmission of sound at theinterface between transducers and sample is considered in advance, whichmeans the acoustic-structural coupled analysis will become thestructural transient analysis. Another reason doing the transition isthat only transmitted data, not the propagation in the air, is useful.At this point, the transmission coefficient should be updated at eachtry in orthogonal array during FE simulating. The wave transmitted intosample is the incident wave multiplied by the transmission coefficient.

Taking the 22.4 Pa detected by the sound receiver as the optimizingobject, after completing all simulation listed in orthogonal array andaccording to range analysis, optimum values can be read out directlyfrom FIG. 13 in which x-coordinate is the level number for each factor,and the vertical is the objective average of a level. The lowest pointin FIG. 13 is the optimum value for each factor: modulus of 1.6 MPa,Poisson ration of 0.30 and density of 1050˜1150 kg/m3 for matrix, and2.6 MPa, 0.39 and 1600˜1700 kg/m3 for inclusion. FIG. 14 shows the FEpressure output of the set of optimum values (1.5 MPa, 0.3 and 1050kg/m3 for matrix, and 2.6 MPa, 0.39 and 1600 kg/m3 for inclusion), whichillustrates that the FE peak value, 24.7 Pa, matches with the objective,22.4 Pa.

The final step includes verification of optimum values. The materialproperties of matrix and inclusion has been optimized out. Verificationis necessary to see if the optimization is real. Some measurements aredone for measuring the density and moduli of matrix and inclusion. Themeasured density is 1050 kg/m3 for matrix, 1690 kg/m3 for inclusion. Themodulus is measured by uniaxial compression test. The samples are shownin FIG. 15. The compression speed is 0.01 mm/s until the strain is up to10%. Five tests are conducted for each sample, and then follows bylinear fit on stress-strain curves as shown in FIG. 16. The slopes ofthese fits is the modulus: 2.1 MPa for matrix and 2.6 MPa for inclusion.It can be concluded that the optimization agrees well with the measuredvalues. Thus, the present invention has been validated by FE simulationand experiments.

SIMULATION EXAMPLE 2

The following is another non-limiting example of a numerical simulationon real multiphase soft tissues. It is to be understood that the exampledescribed herein is presented for illustrative purposes, and is notintended to limit the invention in any way. Equivalents or substitutesare within the scope of the invention.

Numerical model creation and simulation were conducted on the commercialFEM package, Marc Mentat 2018.0.0 (64 bit) (MSC Software Corporation),with the assist of a Python script performing the factorial design, theparameter update and mathematical process. A brain slice was segmentedinto four regions and simulated as shown in FIG. 17. The model wasmeshed into 11261 four-node quadrilateral elements. The incident wave isthe 50 Hz sine signal applied at four positions on the outer boundarydenoted by arrows. As denoted by the black sold dots, twelve locationswere chosen to detect the transmission with the sample rate of 400/s.The acquisition duration is 0.2 s. All tissues were treated as linearelasticity for demonstrating the integrated method first on linearmaterials. The set of shear moduli, [C=1.43, CR=0.66, CC=0.35, BG=0.70]kPa, was used as the objective values and the corresponding output at 12locations as the objective signals. Accordingly, the eleastic moduli are[4.26, 1.97, 1.04, 2.09] kPa if Poisson's ratio of 0.49 is assumed forall tissues. The density is assumed 1000 kg/m³. In this case, fullfactorial design was employed to recognize the optimality for all levelsof factors. Hence, there are four factors corresponding to elasticmoduli of C, CR, CC and BG.

In terms of the reseasonable ranges of their moduli, 3.0 kPa˜6.0 kPa forregion C and 0.1 kPa˜3.0 kPa for other three regions, each factor isassigned eight levels at first, [3.0, 3.4, 3.8, 4.2, 4.6, 5.0, 5.4, 6.0]kPa for C and [0.1, 0.5, 0.9, 1.3, 1.7, 2.1, 2.5, 3.0] for others.Totally, it is 8⁴ (4096) trials for possible combinations of all levels.Running on a PC with Intel® Core™ i7-3770 CPU @ 3.40 GHz and 16.0 GBRAM, each trial takes 21 seconds. Because cross-correlation can be usedto observe the similarity of two signals, the objective function isestablished based on the cross-correlation of the objective signals andeach trial ones. The maximum of cross-correlation is at the zero lagtime if two signals are identical, which is called auto-correlation. Forthis case, the objective function of the optimizatioin problem isdefined as:

$\begin{matrix}{{{\min\;{f(E)}} = \sqrt{\sum\limits_{i = 1}^{12}\left( {1 - \frac{D_{C}^{i}❘_{t = 0}}{D_{A}^{i}❘_{t = 0}}} \right)^{2}}}{{{Subject}\mspace{14mu}{to}\text{:}\mspace{14mu}\min\;{f(E)}} \leq {1.0\%}}} & (9)\end{matrix}$where E=(E_(C), E_(CR), E_(CC), E_(BG)), standing for the elastic moduliattempt in each level, D_(A) ^(i)|_(t=0) is the auto-correlation of themeasured signals of point i at the zero lag time, and D_(C) ^(i)|_(t=0)is the cross-correlation of the trial signals and the measured signalsof point i at the zero lag time. The effect of E on D_(C) ^(i)|_(t=0) isimplicit in equation (9). But, they are bridged in FEM in which E arethe material parameter input and D_(C) ^(i)|_(t=0) is determinedaccording to FEM output. The measured signals that are the objectivesignals come from the FEM simulation with the objective elastic moduliof [4.26, 1.97, 1.04, 2.09] kPa. Applying equation (9) to 4096 trials,min f (E) is reached at [4.2, 2.1, 0.9, 2.1] kPa of the 2732^(nd) trial,meaning that the 2732^(nd) one is the closest to real values among alltrials. The error of the trial is 4.3%, larger than 1.0%, which can besufficient for mapping elasticity. As an illustration for three surfacelocations, the objective signals of the displacement amplitude and thecorresponding simulation signals of the best trial 2732 are plotted inFIG. 18A. In contrast, there is discrepancy for some locations betweentwo sets of signals, which causes the 4.3% error.

To present a more accurate solution, the inverse analysis is continueduntil a min f(E)≤1.0% is achieved. The first round helps to compress thevalue ranges for the later rounds. As a result, based on each value in[4.2, 2.1, 0.9, 2.1] kPa of the first-round, the next round factorialdesign with the smaller step sets [3.90, 4.05, 4.20, 4.35, 4.50] for C,[1.80, 1.95, 2.10, 2.25, 2.40] for CR, [0.60, 0.75, 0.90, 1.05, 1.20]for CC and [1.80, 1.95, 2.10, 2.25, 2.40] for BG, which introduces 625trials. This round ends up with [4.20, 2.10, 1.05, 2.10]. Because theresult is almost the same as that of the first-round, the step of levelsis needed to be further decreased. Levels of the third round are set to[4.10, 4.15, 4.20, 4.25, 4.30] for C, [2.00, 2.05, 2.10, 2.15, 2.20] forCR, [0.80, 0.85, 0.90, 0.95, 1.00] for CC and [1.85, 1.90, 1.95, 2.00,2.05] for BG. After all trials are completed, the third round gives[4.25, 2.00, 1.00, 2.05] with 4.1%. Once more, the level range of thefourth round is narrowed down with the much smaller step, [4.23, 4.25,4.27] for C, [1.98, 2.00, 2.02] for CR, [0.98, 1.00, 1.02] for CC and[2.03, 2.05, 2.07] for BG. Eventually, [4.27, 1.98, 1.02, 2.07] isfinalized as the optimum trial with 1.0%. The comparison of theobjective signals and the corresponding ones of the best trial isplotted in FIG. 18B for the same locations as in FIG. 18A. The two setof signals are almost overlapping, which illustrates again [4.27, 1.98,1.02, 2.07] are the optimum values, consistent with real elastic moduliof [4.26, 1.97, 1.04, 2.09] with errors 0.2%, 0.5%, 1.9% and 1.0%,respectively.

During the entire course, any internal information is neither detectednor processed to map the moduli. Instead, the present inventiondemonstrates that all information for the mapping can be based onsignals on the 12 surface locations. Without wishing to limit thepresent invention to a particular theory or mechanism, the mechanicalresponse of the sample can be determined without relying on responseacquisitions or other invasive procedures within the sample, as themechanical properties of the inclusion can be deduced from analysisperformed on parameters of the surface of the sample.

As used herein, the term “about” refers to plus or minus 10% of thereferenced number.

Various modifications of the invention, in addition to those describedherein, will be apparent to those skilled in the art from the foregoingdescription. Such modifications are also intended to fall within thescope of the appended claims. Each reference cited in the presentapplication is incorporated herein by reference in its entirety.

Although there has been shown and described the preferred embodiment ofthe present invention, it will be readily apparent to those skilled inthe art that modifications may be made thereto which do not exceed thescope of the appended claims. Therefore, the scope of the invention isonly to be limited by the following claims. In some embodiments, thefigures presented in this patent application are drawn to scale,including the angles, ratios of dimensions, etc. In some embodiments,the figures are representative only and the claims are not limited bythe dimensions of the figures. In some embodiments, descriptions of theinventions described herein using the phrase “comprising” includesembodiments that could be described as “consisting of”, and as such thewritten description requirement for claiming one or more embodiments ofthe present invention using the phrase “consisting of” is met.

The reference numbers recited in the below claims are solely for ease ofexamination of this patent application, and are exemplary, and are notintended in any way to limit the scope of the claims to the particularfeatures having the corresponding reference numbers in the drawings.

What is claimed is:
 1. An integrated elastography system (100) forestimating mechanical properties of a sample (110), the systemcomprising: a) an x-ray computed tomography (“CT”) system (101)comprising an x-ray source (102) and a detector (108), wherein the x-raysource emits x-rays towards the sample (110) and wherein the detector(108) collects x-rays emitted from the sample (110); b) a sound wavesystem (103) having one or more acoustic transducers (112) and one ormore receivers (114) configured to be positioned around an outer surfaceof the sample (110), wherein the one or more transducers (112) areconfigured to generate sound waves that impinge on the sample (110), andwherein the one or more receivers (114) receive a first set oftransmitted signals comprising sound waves that transmit through thesample (110); and c) a controller (120) operably coupled to the x-ray CTsystem (101) and the sound wave system (103), wherein the controller(120) has a memory that stores computer readable instructions that, whenexecuted by the controller, causes the controller to: i) generate onlyone x-ray CT image of the sample using the detected x-rays from thesample, and select a matrix and an inclusion based on geometriesdetected from the CT image of the sample; ii) generate the first set oftransmitted signals based on the CT image using the sound wave system(103); iii) select an initial set of values comprising one or more of anelasticity modulus, a Poisson ratio, and a density for the matrix andthe inclusion based on the selection; and iv) perform and repeat aforward analysis sequence to simulate a sound wave transmission throughthe matrix and the inclusion to deduce a mechanical response of thematrix and the inclusion, wherein the forward analysis sequenceoptimizes the initial set of values until the set of signals generatedconverges with the first set of transmitted signals; wherein themechanical response is determined based on the set of signals generated,thereby decoding the mechanical response of the sample without requiringresponse acquisitions from inside the sample.
 2. The system of claim 1,wherein the forward analysis sequence comprises constructing a finiteelement (“FE”) model by generating a mesh of each of the matrix and ofthe inclusion from the CT image, wherein the mesh is generated using anapproximation of the geometries of the matrix and the inclusion, andwherein generation of the mesh includes dividing the CT image intosmaller finite elements.
 3. The system of claim 1, wherein the memoryincludes additional instructions that, when executed by the controller,cause the controller to perform an inverse analysis on the set ofsignals to deduce a modulus of each of the matrix and the inclusion. 4.The system of claim 3, wherein the inverse analysis comprises a waveletsynchro-squeezed transform (“WSST”) comprising: a) generating acontinuous wavelet transform from the set of signals; b) extractinginstantaneous frequency from the continuous wavelet transform; c)reconstructing transmitted signals from continuous wavelet transform andthe frequency; d) decoding the reconstructed transmitted signals intoshear signals and pressure signals; and e) estimating the modulus of thesample based on one or more of the shear signals and the pressuresignals, thereby decoding the mechanical property of the sample usingthe x-ray CT image and the transmitted signals, wherein the estimationis based on a time reversal theory, wherein the modulus is estimatedfrom the shear signals and the pressure signals.
 5. The system of claim1, wherein the one or more transducers (112) are disposed on a surfaceof an enclosure of the integrated elastography system (100) and whereinthe one or more receivers (114) are in contact with outer surface of thesample (110).
 6. The system of claim 1, wherein the sound waves comprisea sine tone burst.
 7. The system of claim 1, wherein the initial set ofvalues is selected based on an estimate of the elasticity modulus, thePoisson ratio, and the density for the matrix and the inclusion.
 8. Thesystem of claim 1, wherein the mechanical response includes one or moreof displacement, velocity, acceleration, and a pressure within thesample.
 9. The system of claim 1, wherein the sample is a biologicalsample, wherein the matrix comprises tissue and the inclusion comprisenon-tissue biological material.
 10. The integrated elastography systemof claim 9, wherein the biological material is a tumor or foreign mass.11. The integrated elastography system of claim 1, wherein the sample isa composite material comprising particles embedded in a solid medium.12. An integrated elastography system (100) comprising: a) an x-raycomputed tomography (“CT”) system (101) comprising an x-ray source (102)and a detector (108), wherein the x-ray source emits x-rays towards asample (110) and wherein the detector (108) collects x-rays emitted fromthe sample (110); b) a sound wave system (103) having a plurality ofacoustic transducers (112) and a plurality of receivers (114) configuredto be positioned at or near an outer surface of the sample (110),wherein the plurality of transducers (112) are configured to generatesound waves that impinge on the sample (110), and wherein the pluralityof the receivers (114) receive a first set of transmitted signals,wherein the first set of transmitted signals include the sound wavesthat transmit through the sample (110); and c) a controller (120)operably coupled to the x-ray CT system (101) and the sound wave system(103), wherein the controller (120) has memory that stores computerreadable instructions that, when executed by the controller, causes thecontroller to: i. generate only one x-ray CT image of the sample usingthe detected x-rays from the sample, and select a matrix and aninclusion based on geometries detected from the CT image of the sample;ii. generate a mesh of each of the matrix and of the inclusion from theCT image, wherein the mesh is generated using an approximation of thegeometries of the matrix and the inclusion, and wherein generation ofthe mesh includes dividing the CT image into smaller finite elements;iii. choose an initial set of values comprising one or more of anelasticity modulus, a Poisson ratio, and a density for the matrix andthe inclusion based on the selection; iv. perform a finite elementanalysis on the CT image using the set of initial values of anelasticity modulus, a Poisson ratio, and a density of the sample togenerate a second set of signals; v. perform a wavelet synchro-squeezedtransform (“WSST”) on a deformation of shear and pressure wavesoutputted from the finite element analysis to generate a continuouswavelet transform; vi. extract an instantaneous frequency from thecontinuous wavelet transform; vii. reconstruct the transmitted signalsfrom the continuous wavelet transform and the frequency; viii. decodethe reconstructed transmitted signals into shear signals and pressuresignals; ix. repeat step (iv)-(viii) to optimize the set of initialvalues and generate subsequent set of signals until the set of signalsconverge with the first set of transmitted signals; and x. estimate anelastic modulus of the sample based on one or more of the shear signalsand the pressure signals, wherein the estimation is based on a timereversal theory, and wherein a mechanical property of the sample isestimated from the elastic modulus, thereby decoding the mechanicalproperty of the sample using the x-ray CT image and the transmittedsignals without relying on response acquisitions from inside the sample.13. The integrated elastography system of claim 12, wherein the timereversal theory includes generating a time reversal displacement fieldand a time reversal strain field, wherein the elastic modulus isdetermined from a ratio of the time reversal displacement field and thetime reversal strain field.
 14. The integrated elastography system ofclaim 12, wherein the sound waves comprise a sine tone burst.
 15. Theintegrated elastography system of claim 12, wherein the initial set ofvalues are chosen based on an estimate of the elasticity modulus, thePoisson ratio, and the density for the matrix and the inclusion.
 16. Amethod for determining a modulus of a sample, the method comprising: (a)obtaining only one x-ray computed tomography (“CT”) image of the sampleusing x-rays collected from the sample, wherein the x-rays are collectedusing an x-ray CT system (101) comprising an x-ray source (102) and adetector (108), wherein the x-ray source emits x-rays towards the sample(110) and the detector (108) collects x-rays emitted from the sample(110); (b) positioning at or near an outer surface of the sample (110) asound wave system (103) comprising one or more acoustic transducers(112) and one or more receivers (114), wherein the one or moretransducers (112) generate sound waves that impinge on the sample (110),and the one or more receivers (114) receive a first set of transmittedsignals; (c) impinging the sample with sound waves generated from theone or more transducers (112), wherein the one or more receivers (114)receive the first set of transmitted signals comprising the sound wavesthat transmit through the sample (110); (d) performing a finite elementanalysis on the CT image using a set of initial values of an elasticitymodulus, a Poisson ratio, and a density of the sample to generate asecond set of signals; (e) performing a cycle operation of optimizingthe set of initial values to generate subsequent sets of signals untilthe set of signals converges with the first set of transmitted signals;and (f) determining a mechanical response of the sample based on theoptimized set of initial values, wherein the mechanical responseincludes one or more of a displacement, a velocity, an acceleration, anda pressure of the sample.
 17. The method of claim 16, further comprisingperforming an inverse analysis on the set of signals generated to deducea modulus of the sample.
 18. The method of claim 17, wherein the inverseanalysis comprises a wavelet synchro-squeezed transform (“WSST”)comprising: (a) generating a continuous wavelet transform from the setof signals; (b) extracting instantaneous frequency from the continuouswavelet transform; (c) reconstructing transmitted signals fromcontinuous wavelet transform and the frequency; (d) decoding thereconstructed transmitted signals into shear signals and pressuresignals; and (e) estimate the elastic modulus of the sample based on oneor more of the shear signals and the pressure signals, wherein theestimation is based on a time reversal theory, and wherein the elasticmodulus is estimated from the shear signals and the pressure signals,thereby decoding mechanical property of the sample using the x-ray CTimage and the transmitted signals.
 19. The method of claim 16, whereinperforming the finite element analysis on the CT image comprises: (a)selecting a matrix and an inclusion based on geometries detected fromthe CT image of the sample; and (b) generating a mesh of each of thematrix and of the inclusion, wherein the mesh is generated using anapproximation of the geometries of the matrix and the inclusion from theCT image, and wherein generation of the mesh includes dividing the CTimage into smaller finite elements.
 20. The method of claim 19, furthercomprising determining the mechanical response and the modulus of eachof the matrix and the inclusion.